Solving the System of Equations
In this article, we will solve the system of equations:
$\frac{1}{2x} - \frac{1}{y} = -1$
$\frac{1}{x} + \frac{1}{2y} = 8$
Step 1: Simplify the Equations
Let's simplify the equations by multiplying both sides of each equation by the least common multiple of the denominators.
For the first equation, the least common multiple of 2x and y is 2xy. So, we multiply both sides by 2xy:
$y - 2x = -2xy$
For the second equation, the least common multiple of x and 2y is 2xy. So, we multiply both sides by 2xy:
$2y + x = 16xy$
Step 2: Solve for x and y
Now, we have two equations with two variables:
$y - 2x = -2xy$ $2y + x = 16xy$
We can solve for x by isolating x in the second equation:
$x = 16xy - 2y$
Substitute this expression for x into the first equation:
$y - 2(16xy - 2y) = -2xy$
Simplify the equation:
$y - 32xy + 4y = -2xy$
Combine like terms:
$-32xy + 5y = -2xy$
Now, divide both sides by -2xy:
$16x - y = 1$
Substitute this expression for y into one of the original equations:
$x = 16x(16x - (1 + 16x))$
Simplify the equation:
$x = \frac{1}{15}$
Now that we have the value of x, we can find the value of y:
$y = 16(\frac{1}{15})(\frac{1}{15} - 1)$
Simplify the equation:
$y = 14$
Solution
The solution to the system of equations is x = 1/15 and y = 14.
Therefore, the values of x and y that satisfy both equations are x = 1/15 and y = 14.
